PROBLEM:
There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges.  The boxes have been incorrectly labeled such that no label identifies the actual contents of the box it labels.  Opening just one box, and without looking in the box, you take out one piece of fruit.  By looking at the fruit, how can you immediately label all of the boxes correctly.

Which box did you open and how can you be sure to label all boxes correctly?
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
SOLUTION: 
The box that must be opened is the one labeled "apples and oranges."  By definition, whichever fruit is inside, is the only fruit type that that box contains.  Let's say that you found an apple in that box that was labeled with both apples and oranges; because you know it must therefore only contain apples, then you conclude that the box that is labeled "oranges" cannot contain only oranges, as all boxes have been said to be mislabeled.  Thus, the box labeled "oranges" must contain both apples and oranges, leaving the box labeled "apples" to contain only oranges.